Understanding Compound Interest vs. Simple Interest

This question asks for the difference between the compound interest (CI) and simple interest (SI) earned on a principal amount over a specific period at a given interest rate. Let's break down how to calculate both and find the difference.

The principal amount (P) is Rs. 8,000.

The time period (T) is 2 years.

The rate of interest (R) is 5% per annum (p.a.).

Calculating Simple Interest (SI)

Simple Interest is calculated only on the original principal amount for the entire duration. The formula for Simple Interest is:

$\text{SI} = \frac{P \times R \times T}{100}$

Plugging in the given values:

$\text{SI} = \frac{8000 \times 5 \times 2}{100}$

$\text{SI} = \frac{80000}{100}$

$\text{SI} = 800$

So, the Simple Interest for 2 years is Rs. 800.

Calculating Compound Interest (CI)

Compound Interest is calculated on the principal amount as well as on the accumulated interest from previous periods. The formula for the Amount (A) after T years when compounded annually is:

$A = P \left(1 + \frac{R}{100}\right)^T$

Plugging in the given values:

$A = 8000 \left(1 + \frac{5}{100}\right)^2$

$A = 8000 \left(1 + 0.05\right)^2$

$A = 8000 \left(1.05\right)^2$

$A = 8000 \times 1.1025$

$A = 8820$

The amount after 2 years is Rs. 8,820.

Compound Interest (CI) is the total amount minus the principal amount:

$\text{CI} = A - P$

$\text{CI} = 8820 - 8000$

$\text{CI} = 820$

So, the Compound Interest for 2 years is Rs. 820.

Finding the Difference between CI and SI

The difference between the Compound Interest and the Simple Interest is:

Difference = CI - SI

Difference = $820 - 800$

Difference = $20$

The difference between the compound interest and the simple interest is Rs. 20.

Alternative Method: Direct Formula for CI - SI Difference for 2 Years

For a period of 2 years, the difference between Compound Interest and Simple Interest can also be calculated directly using the formula:

Difference $= P \left(\frac{R}{100}\right)^2$

Plugging in the values:

Difference $= 8000 \left(\frac{5}{100}\right)^2$

Difference $= 8000 \left(0.05\right)^2$

Difference $= 8000 \times 0.0025$

Difference $= 20$

This confirms the previous calculation. The difference is indeed Rs. 20.

The difference between the compound interest and the simple interest on Rs. 8,000 for 2 years at 5% p.a. is Rs. 20.

Revision Table: Key Concepts

Additional Information: Factors Affecting Interest Difference

The difference between compound interest and simple interest depends on several factors:

• Time Period: The difference increases significantly as the time period increases. For a period of only one year, CI and SI are the same if the compounding is annual. The difference starts appearing from the second year onwards.
• Rate of Interest: A higher interest rate leads to a larger difference between CI and SI, assuming the principal and time are constant.
• Principal Amount: A larger principal amount will result in a larger difference, assuming the rate and time are constant.
• Compounding Frequency: If interest is compounded more frequently than annually (e.g., half-yearly, quarterly, monthly), the compound interest will be higher, and thus the difference between CI and SI will also be larger. In this problem, it's assumed to be compounded annually since no other frequency is mentioned.

Understanding the difference between these two types of interest is crucial for financial planning, investments, and loans.